In a categorification of tensor products of fundamental representations of quantum sl(k) via highest weight categories, the indecomposable tilting modules descend to the canonical basis. Since projective functors map tilting modules to tilting modules, the coefficients of the canonical basis of tensor products of finite dimensional, irreducible representations under the action of the Chevalley generators are positive.
We show the positivity of the canonical basis for a modified quantum affine $\mathfrak{sl}_n$ under the comultiplication. Moreover, we establish the positivity of the i-canonical basis in [LW15] with respect to the coideal subalgebra structure.
For the quantum symmetric pair $(\textbf{U}, \textbf{U}^{\imath})$ of type AIII/AIV, we show various positivity properties of the $\imath$-canonical bases on finite-dimensional simple $\textbf{U}$-modules, as well as their tensor product.
The positivity property for canonical bases asserts that the structure constants of the multiplication for the canonical basis are in ${\mathbb N}[v,v^{-1}]$. Let $\mathbf U$ be the quantum group over ${\mathbb Q}(v)$ associated with a symmetric Cartan datum. The positivity property for the positive part ${\mathbf U}^+$ of ${\mathbf U}$ was proved by Lusztig. He conjectured that the positivity property holds for the modified form $\dot{\mathbf U}$ of ${\mathbf U}$. In this paper, we prove that the structure constants for the canonical basis of $\dot{\mathbf U}(\widehat{\frak{sl}}_n)$ coincide …
We provide examples for negativity of structure constants of the stably canonical basis of modified quantum $\mathfrak{gl}_n$ and an analogous basis of modified quantum coideal algebra of $\mathfrak{gl}_n$. In contrast, we construct the canonical basis of the modified quantum coideal algebra of $\mathfrak{sl}_n$, establish the positivity of its structure constants, the positivity with respect to a geometric bilinear form as well as the positivity of its action on the tensor powers of the natural representation. The matrix coefficients of the transfer map on these Schur algebras with respect to…
An algorithm is described to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for modules that are constructed as a submodule of a tensor product of modules with known canonical bases.
In the present article, a basis of the coordinate algebra of the multi-parameter quantized matrix is constructed by using an elementary method due to Lusztig. The construction depends heavily on an anti-automorphism, the bar action. The exponential nature of the bar action is derived which provides an inductive way to compute the basis elements. By embedding the basis into the dual basis of Lusztig's canonical basis of $U_q(n^-)$, the positivity properties of the basis as well as the positivity properties of the canonical basis of the modified quantum enveloping algebra of type $A$, which has …
It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Geiß, Leclerc and Schröer has recently given a parametrization of irreducible components of nilpotent varieties in studying cluster algebras. In this paper we show that their parametrization coincides with Lusztig's parametrization of the canonical basis.
We show the existence of a finite group $G$ having an irreducible character $χ$ with Frobenius-Schur indicator $ν_2(χ){=}{+}1$ such that $χ^2$ has an irreducible constituent $φ$ with $ν_2(φ){=}{-}1$. This provides counterexamples to the positivity conjecture in rational CFT and a conjecture of Zhenghan Wang about pivotal fusion categories.
It has been known since their birth in Bao and Wang's work that the $\imath$canonical bases of $\imath$quantum groups are not stable in general. In the author's previous work, the stability of $\imath$canonical bases of certain quasi-split types turned out to be closely related to the theory of $\imath$crystals. In this paper, we prove the stability of $\imath$canonical bases of irreducible finite type of real rank $1$, for which the theory of $\imath$crystals has not been developed, by means of global and local crystal bases.
We show that tensoring the Laplace and Dolbeault-Dirac operators of a Kähler structure (with closed integral) by a negative Hermitian holomorphic module, produces operators with spectral gaps around zero. The proof is based on the recently established Akizuki-Nakano identity of a noncommutative Kähler structure. This general framework is then applied to the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds, and it is shown that twisting their Dirac and Laplace operators by negative line bundles produces a spectral gap, for q sufficiently close to 1. The main technical challen…
The aim of this paper is to study the canonical filtration $L(λ)_l$ of an irreducible finite dimensional $\operatorname{SL}(V)$-module $L(λ)$ using the universal enveloping algebra $U(\mathfrak{sl}(V))$ and the annihilator ideal $ann(v)$ of a highest weight vector $v$ in $L(λ)$. We give a basis for $L(λ)_l$ and calculate the dimension of $L(λ)_l$ as a function of $l$. This is done in terms of the universal enveloping algebra of the nilpotent radical of an opposite parabolic sub algebra of the stabilizer Lie algebra of a flag $V_*$ in $V$ with respect to a choice of roots for $\mathfrak{sl}(V)$…
Let $n$ be the maximal nilpotent subalgebra of a simple complex Lie algebra $g$. We introduce the notion of imaginary vector in the dual canonical basis of $U_q(n)$, and we give examples of such vectors for types $A_n (n\ge 5)$, $B_n (n\ge 3)$, $C_n (n\ge 3)$, $D_n (n\ge 4)$, and all exceptional types. This disproves a conjecture of Berenstein and Zelevinsky about $q$-commuting products of vectors of the dual canonical basis. It also shows the existence of finite-dimensional irreducible representations of quantum affine algebras whose tensor square is not irreducible.
Given any symmetric Cartan datum, Lusztig has provided a pair of key lemmas to construct the perverse sheaves over the corresponding quiver and the functions of irreducible components over the corresponding preprojective algebra respectively. In the present article, we prove that these two inductive algorithms of Lusztig coincide. Consequently we can define two colored graphs and prove that they are isomorhic. This result finishes the statement that Lusztig's functions of irreducible components are basis of the enveloping algebra and deduces the crystal structure (in the sense of Kashiwara-Sai…
We give a geometric interpretation of the inner product on the modified quantum group of $\hat{\mathfrak{sl}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new geometric construction of the canonical basis.
In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed ${\rm PGL}_2$-local systems on a punctured surface $S$. The moduli space is birational to a cluster $\mathcal{X}$-variety, whose positive real points recover the enhanced Teichmüller space of $S$. Their basis is enumerated by integral laminations on $S$, which are collections of closed curves in $S$ with integer weights. Around ten years later, a quantized version of this basis, still enumerated by integral laminations, was constructed by Allegretti and Kim. For each choice of an…
We give a fast algorithm for computing the canonical basis of an irreducible highest-weight module for $U_q(\hat{\mathfrak{sl}}_e)$, generalising the LLT algorithm.
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
We give closed formulas for all vectors of the canonical basis of a level 2 irreducible integrable representation of $U_v(sl_\infty)$. These formulas coincide at v=1 with Lusztig's formulas for the constructible characters of the Iwahori-Hecke algebras of type B and D.
Using the isomorphism between highest weight U_q(sl_2)-modules and homologies of certain local systems on the configuration spaces, constructed by Varchenko, we give a geometric construction of the dual of the Lusztig's canonical basis in a tensor product of irreducible finite-dimensional U_q(sl_2)-modules.
In the basic representation of $U_q(\hat{sl}(2))$ realized via the algebra of symmetric functions we compare the canonical basis with the basis of Macdonald polynomials with $q=t^2$. We show that the Macdonald polynomials are invariant with respect to the bar involution defined abstractly on the representations of quantum groups. We also prove that the Macdonald scalar product coincides with the abstract Kashiwara form. This implies, in particular, that the Macdonald polynomials form an intermediate basis between the canonical basis and the dual canonical basis, and the coefficients of the tra…
Noncommutative Kähler structures were recently introduced by the second author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic one-forms) we provide simple cohomological criteria for positivity, offering a means to avoid explicit curvature calculations. These general results a…
